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Betti tables of monomial ideals fixed by permutations of the variables
Transactions of the American Mathematical Society ( IF 1.2 ) Pub Date : 2020-08-05 , DOI: 10.1090/tran/8159
Satoshi Murai

Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of $I_n$ for all large intergers $n$ from the $\mathbb Z^m$-graded Betti table of $I_m$ for some integer $m$. Our main result shows that the projective dimension and the regularity of $I_n$ eventually become linear functions on $n$, confirming a special case of conjectures posed by Le, Nagel, Nguyen and R\"omer.

中文翻译:

由变量排列固定的单项式理想 Betti 表

令 $S_n$ 是一个多项式环,在一个域上有 $n$ 个变量,$\{I_n\}_{n \geq 1}$ 是一个理想链,使得每个 $I_n$ 都是 $S_n$ 固定的单项式理想通过变量的排列。在本文中,我们提出了一种方法来确定所有大整数 $n$ 的 $I_n$ 的 Betti 表的所有非零位置,从 $\mathbb Z^m$ 分级的 $I_m$ 的 Betti 表中的某个整数 $m$ . 我们的主要结果表明,$I_n$ 的射影维数和正则性最终成为 $n$ 上的线性函数,证实了 Le、Nagel、Nguyen 和 R\"omer 提出的猜想的特例。
更新日期:2020-08-05
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